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Anti-classification results for weakly mixing diffeomorphisms 弱混合衍射的反分类结果
IF 1.4 2区 数学
Mathematische Annalen Pub Date : 2024-05-22 DOI: 10.1007/s00208-024-02890-0
Philipp Kunde
{"title":"Anti-classification results for weakly mixing diffeomorphisms","authors":"Philipp Kunde","doi":"10.1007/s00208-024-02890-0","DOIUrl":"https://doi.org/10.1007/s00208-024-02890-0","url":null,"abstract":"<p>We extend anti-classification results in ergodic theory to the collection of weakly mixing systems by proving that the isomorphism relation as well as the Kakutani equivalence relation of weakly mixing invertible measure-preserving transformations are not Borel sets. This shows in a precise way that classification of weakly mixing systems up to isomorphism or Kakutani equivalence is impossible in terms of computable invariants, even with a very inclusive understanding of “computability”. We even obtain these anti-classification results for weakly mixing area-preserving smooth diffeomorphisms on compact surfaces admitting a non-trivial circle action as well as real-analytic diffeomorphisms on the 2-torus.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"35 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141149068","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Classification of radial blow-up at the first critical exponent for the Lin–Ni–Takagi problem in the ball 球内林-尼-高木问题在第一个临界指数处的径向膨胀分类
IF 1.4 2区 数学
Mathematische Annalen Pub Date : 2024-05-18 DOI: 10.1007/s00208-024-02888-8
Denis Bonheure, Jean-Baptiste Casteras, Bruno Premoselli
{"title":"Classification of radial blow-up at the first critical exponent for the Lin–Ni–Takagi problem in the ball","authors":"Denis Bonheure, Jean-Baptiste Casteras, Bruno Premoselli","doi":"10.1007/s00208-024-02888-8","DOIUrl":"https://doi.org/10.1007/s00208-024-02888-8","url":null,"abstract":"<p>We investigate the behaviour of radial solutions to the Lin–Ni–Takagi problem in the ball <span>(B_R subset mathbb {R}^N)</span> for <span>(N ge 3)</span>: </p><span>$$begin{aligned} left{ begin{array}{ll} - triangle u_p + u_p = |u_p|^{p-2}u_p &amp;{}quad text { in } B_R, partial _nu u_p = 0 &amp;{}quad text { on } partial B_R, end{array} right. end{aligned}$$</span><p>when <i>p</i> is close to the first critical Sobolev exponent <span>(2^* = frac{2N}{N-2})</span>. We obtain a complete classification of finite energy radial smooth blowing up solutions to this problem. We describe the conditions preventing blow-up as <span>(p rightarrow 2^*)</span>, we give the necessary conditions in order for blow-up to occur and we establish their sharpness by constructing examples of blowing up sequences. Our approach allows for asymptotically supercritical values of <i>p</i>. We show in particular that, if <span>(p ge 2^*)</span>, finite-energy radial solutions are precompact in <span>(C^2(overline{B_R}))</span> provided that <span>(Nge 7)</span>. Sufficient conditions are also given in smaller dimensions if <span>(p=2^*)</span>. Finally we compare and interpret our results in light of the bifurcation analysis of Bonheure, Grumiau and Troestler in (Nonlinear Anal 147:236–273, 2016).</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"9 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141063454","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Compressible fluids interacting with 3D visco-elastic bulk solids 与三维粘弹性块状固体相互作用的可压缩流体
IF 1.4 2区 数学
Mathematische Annalen Pub Date : 2024-05-14 DOI: 10.1007/s00208-024-02886-w
Dominic Breit, Malte Kampschulte, Sebastian Schwarzacher
{"title":"Compressible fluids interacting with 3D visco-elastic bulk solids","authors":"Dominic Breit, Malte Kampschulte, Sebastian Schwarzacher","doi":"10.1007/s00208-024-02886-w","DOIUrl":"https://doi.org/10.1007/s00208-024-02886-w","url":null,"abstract":"<p>We consider the physical setup of a three-dimensional fluid–structure interaction problem. A viscous compressible gas or liquid interacts with a nonlinear, visco-elastic, three-dimensional bulk solid. The latter is described by an evolution with inertia, a non-linear dissipation term and a term that relates to a non-convex elastic energy functional. The fluid is modelled by the compressible Navier–Stokes equations with a barotropic pressure law. Due to the motion of the solid, the fluid domain is time-changing. Our main result is the long-time existence of a weak solution to the coupled system until the time of a collision. The nonlinear coupling between the motions of the two different matters is established via the method of minimising movements. The motion of both the solid and the fluid is chosen via an incrimental minimization with respect to dissipative and static potentials. These variational choices together with a careful construction of an underlying flow map for our approximation then directly result in the pressure gradient and the material time derivatives.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"189 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140936342","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Rabinowitz Floer homology for prequantization bundles and Floer Gysin sequence 预量化束的拉比诺维兹-弗洛尔同源性和弗洛尔-吉辛序列
IF 1.4 2区 数学
Mathematische Annalen Pub Date : 2024-05-09 DOI: 10.1007/s00208-024-02878-w
Joonghyun Bae, Jungsoo Kang, Sungho Kim
{"title":"Rabinowitz Floer homology for prequantization bundles and Floer Gysin sequence","authors":"Joonghyun Bae, Jungsoo Kang, Sungho Kim","doi":"10.1007/s00208-024-02878-w","DOIUrl":"https://doi.org/10.1007/s00208-024-02878-w","url":null,"abstract":"<p>Let <i>Y</i> be a prequantization bundle over a closed spherically monotone symplectic manifold <span>(Sigma )</span>. Adapting an idea due to Diogo and Lisi, we study a split version of Rabinowitz Floer homology for <i>Y</i> in the following two settings. First, <span>(Sigma )</span> is a symplectic hyperplane section of a closed symplectic manifold <i>X</i> satisfying a certain monotonicity condition; in this case, <span>(X {{setminus }} Sigma )</span> is a Liouville filling of <i>Y</i>. Second, the minimal Chern number of <span>(Sigma )</span> is greater than one, which is the case where the Rabinowitz Floer homology of the symplectization <span>(mathbb {R} times Y)</span> is defined. In both cases, we construct a Gysin-type exact sequence connecting the Rabinowitz Floer homology of <span>(X{setminus }Sigma )</span> or <span>(mathbb {R} times Y)</span> and the quantum homology of <span>(Sigma )</span>. As applications, we discuss the invertibility of a symplectic hyperplane section class in quantum homology, the isotopy problem for fibered Dehn twists, the orderability problem for prequantization bundles, and the existence of translated points. We also provide computational results based on the exact sequence that we construct.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"161 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140936314","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Solutions with moving singularities for a one-dimensional nonlinear diffusion equation 一维非线性扩散方程的移动奇点解决方案
IF 1.4 2区 数学
Mathematische Annalen Pub Date : 2024-05-08 DOI: 10.1007/s00208-024-02882-0
Marek Fila, Jin Takahashi, Eiji Yanagida
{"title":"Solutions with moving singularities for a one-dimensional nonlinear diffusion equation","authors":"Marek Fila, Jin Takahashi, Eiji Yanagida","doi":"10.1007/s00208-024-02882-0","DOIUrl":"https://doi.org/10.1007/s00208-024-02882-0","url":null,"abstract":"<p>The aim of this paper is to study singular solutions for a one-dimensional nonlinear diffusion equation. Due to slow diffusion near singular points, there exists a solution with a singularity at a prescribed position depending on time. To study properties of such singular solutions, we define a minimal singular solution as a limit of a sequence of approximate solutions with large Dirichlet data. Applying the comparison principle and the intersection number argument, we discuss the existence and uniqueness of a singular solution for an initial-value problem, the profile near singular points and large-time behavior of solutions. We also give some results concerning the appearance of a burning core, convergence to traveling waves and the existence of an entire solution.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"10 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140936341","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A biharmonic analogue of the Alt–Caffarelli problem 阿尔特-卡法雷利问题的双谐波类似物
IF 1.4 2区 数学
Mathematische Annalen Pub Date : 2024-05-06 DOI: 10.1007/s00208-024-02883-z
Hans-Christoph Grunau, Marius Müller
{"title":"A biharmonic analogue of the Alt–Caffarelli problem","authors":"Hans-Christoph Grunau, Marius Müller","doi":"10.1007/s00208-024-02883-z","DOIUrl":"https://doi.org/10.1007/s00208-024-02883-z","url":null,"abstract":"<p>We study a natural biharmonic analogue of the classical Alt–Caffarelli problem, both under Dirichlet and under Navier boundary conditions. We show existence, basic properties and <span>(C^{1,alpha })</span>-regularity of minimisers. For the Navier problem we also obtain a symmetry result in case that the boundary data are radial. We find this remarkable because the problem under investigation is of higher order. Computing radial minimisers explicitly we find that the obtained regularity is optimal.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"24 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140886782","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Flux and symmetry effects on quantum tunneling 量子隧道的通量和对称效应
IF 1.4 2区 数学
Mathematische Annalen Pub Date : 2024-05-05 DOI: 10.1007/s00208-024-02874-0
Bernard Helffer, Ayman Kachmar, Mikael Persson Sundqvist
{"title":"Flux and symmetry effects on quantum tunneling","authors":"Bernard Helffer, Ayman Kachmar, Mikael Persson Sundqvist","doi":"10.1007/s00208-024-02874-0","DOIUrl":"https://doi.org/10.1007/s00208-024-02874-0","url":null,"abstract":"<p>Motivated by the analysis of the tunneling effect for the magnetic Laplacian, we introduce an abstract framework for the spectral reduction of a self-adjoint operator to a hermitian matrix. We illustrate this framework by three applications, firstly the electro-magnetic Laplacian with constant magnetic field and three equidistant potential wells, secondly a pure constant magnetic field and Neumann boundary condition in a smoothed triangle, and thirdly a magnetic step where the discontinuity line is a smoothed triangle. Flux effects are visible in the three aforementioned settings through the occurrence of eigenvalue crossings. Moreover, in the electro-magnetic Laplacian setting with double well radial potential, we rule out an artificial condition on the distance of the wells and extend the range of validity for the tunneling approximation recently established in Fefferman et al. (SIAM J Math Anal 54: 1105–1130, 2022), Helffer &amp; Kachmar (Pure Appl Anal, 2024), thereby settling the problem of electro-magnetic tunneling under constant magnetic field and a sum of translated radial electric potentials.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"81 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140889703","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
$$L^p$$ bounds for Stein’s spherical maximal operators 斯坦因球面最大算子的 $L^p$$ 边界
IF 1.4 2区 数学
Mathematische Annalen Pub Date : 2024-05-04 DOI: 10.1007/s00208-024-02884-y
Naijia Liu, Minxing Shen, Liang Song, Lixin Yan
{"title":"$$L^p$$ bounds for Stein’s spherical maximal operators","authors":"Naijia Liu, Minxing Shen, Liang Song, Lixin Yan","doi":"10.1007/s00208-024-02884-y","DOIUrl":"https://doi.org/10.1007/s00208-024-02884-y","url":null,"abstract":"<p>Let <span>({mathfrak {M}}^alpha )</span> be the spherical maximal operators of complex order <span>(alpha )</span> on <span>({{mathbb {R}}^n})</span>. In this article we show that when <span>(nge 2)</span>, suppose </p><span>$$begin{aligned} Vert {mathfrak {M}}^{alpha } f Vert _{L^p({{mathbb {R}}^n})} le CVert f Vert _{L^p({{mathbb {R}}^n})} end{aligned}$$</span><p>holds for some <span>(alpha )</span> and <span>(pge 2)</span>, then we must have that <span>(textrm{Re},alpha ge max {1/p-(n-1)/2, -(n-1)/p }.)</span> In particular, when <span>(n=2)</span>, we prove that <span>( Vert {mathfrak {M}}^{alpha } f Vert _{L^p({{mathbb {R}}^2})} le CVert f Vert _{L^p({{mathbb {R}}^2})})</span> if <span>(textrm{Re} ! alpha &gt;max {1/p-1/2, -1/p})</span>, and consequently the range of <span>(alpha )</span> is sharp in the sense that the estimate fails for <span>(textrm{Re} alpha &lt;max {1/p-1/2, -1/ p}.)</span></p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"42 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140886783","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Studying Hilbert’s 10th problem via explicit elliptic curves 通过显式椭圆曲线研究希尔伯特第 10 个问题
IF 1.4 2区 数学
Mathematische Annalen Pub Date : 2024-05-04 DOI: 10.1007/s00208-024-02879-9
Debanjana Kundu, Antonio Lei, Florian Sprung
{"title":"Studying Hilbert’s 10th problem via explicit elliptic curves","authors":"Debanjana Kundu, Antonio Lei, Florian Sprung","doi":"10.1007/s00208-024-02879-9","DOIUrl":"https://doi.org/10.1007/s00208-024-02879-9","url":null,"abstract":"<p>N. García-Fritz and H. Pasten showed that Hilbert’s 10th problem is unsolvable in the ring of integers of number fields of the form <span>(mathbb {Q}(root 3 of {p},sqrt{-q}))</span> for positive proportions of primes <i>p</i> and <i>q</i>. We improve their proportions and extend their results to the case of number fields of the form <span>(mathbb {Q}(root 3 of {p},sqrt{Dq}))</span>, where <i>D</i> belongs to an explicit family of positive square-free integers. We achieve this by using multiple elliptic curves, and replace their Iwasawa theory arguments by a more direct method.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"49 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140889882","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Surface quasi-geostrophic equation perturbed by derivatives of space-time white noise 受时空白噪声导数扰动的表面准地转方程
IF 1.4 2区 数学
Mathematische Annalen Pub Date : 2024-05-03 DOI: 10.1007/s00208-024-02881-1
Martina Hofmanová, Xiaoyutao Luo, Rongchan Zhu, Xiangchan Zhu
{"title":"Surface quasi-geostrophic equation perturbed by derivatives of space-time white noise","authors":"Martina Hofmanová, Xiaoyutao Luo, Rongchan Zhu, Xiangchan Zhu","doi":"10.1007/s00208-024-02881-1","DOIUrl":"https://doi.org/10.1007/s00208-024-02881-1","url":null,"abstract":"<p>We consider a family of singular surface quasi-geostrophic equations </p><span>$$begin{aligned} partial _{t}theta +ucdot nabla theta =-nu (-Delta )^{gamma /2}theta +(-Delta )^{alpha /2}xi ,qquad u=nabla ^{perp }(-Delta )^{-1/2}theta , end{aligned}$$</span><p>on <span>([0,infty )times {mathbb {T}}^{2})</span>, where <span>(nu geqslant 0)</span>, <span>(gamma in [0,3/2))</span>, <span>(alpha in [0,1/4))</span> and <span>(xi )</span> is a space-time white noise. For the first time, we establish the <i>existence of infinitely many non-Gaussian</i></p><ul>\u0000<li>\u0000<p>probabilistically strong solutions for every initial condition in <span>(C^{eta })</span>, <span>(eta &gt;1/2)</span>;</p>\u0000</li>\u0000<li>\u0000<p>ergodic stationary solutions.</p>\u0000</li>\u0000</ul><p> The result presents a single approach applicable in the subcritical, critical as well as supercritical regime in the sense of Hairer (Invent Math 198(2):269–504, 2014). It also applies in the particular setting <span>(alpha =gamma /2)</span> which formally possesses a Gaussian invariant measure. In our proof, we first introduce a modified Da Prato–Debussche trick which, on the one hand, permits to convert irregularity in time into irregularity in space and, on the other hand, increases the regularity of the linear solution. Second, we develop a convex integration iteration for the corresponding nonlinear equation which yields non-unique non-Gaussian solutions satisfying powerful global-in-time estimates and generating stationary as well as ergodic stationary solutions.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"116 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140886773","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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